...there are a good many proverbs which characterize strikingly the typical procedures followed in solving problems, the points of common sense involved, the usual tricks, and the usual errors. ....It would be foolish to regard proverbs as an authoritative source of universally applicabal wisdom but it would be a pity to disregard the graphic description of heuristic procedures provided by proverbs.
It could be an interesting task to collect and group proverbs about planning, seeking means, and choosing between lines of action, in short, proverbs about problem solving......The proverbs quoted will be printed in italics.
1. The very first thing we must do for our problem is to understand it: Who understands ill, answers ill.
We must see clearly the end we have to attain: Think for the end before you begin.
..."respice finem" is the saying in Latin. Unfortunately, not everybody heeds such good advice, and people often start speculating, talking, and even acting fussily without having properly understood the aim for which they should work. A fool looks to the beginning, a wise man regards the end. If the end is not clear in our mind, we may easily stray from the problem and drop it. A wise man begins in the end, a fool ends in the beginning.
.....We have no chance to solve a difficult problem without a strong desire to solve it, but with such desire there is a chance. Where there is a will, there is a way.
2. Devising a plan, conceiving the idea of an appropriate action, is the main achievement in the solution of a problem. A good idea is a piece of good fortune, an inspiration, a gift of the gods, and we have to deserve it: Diligence is the mother of good luck. Perseverance kills the game. An oak is not felled at one stroke. If at first you don't succeed, try, try again. It is not enough however to try repeatedly, we must try different means, vary our trials. Try all the keys in the bunch. Arrows are made of all sorts of wood.
We must adapt our trials to the circumstances. As the wind blows you must set your sail. Cut your coat according to the cloth. We must do as we may if we can't do as we would. If we have failed, we must try something else. A wise man changes his mind, a fool never does. We should even be prepared from the outset for a possible failure of our scheme and have another one in reserve. Have two strings to your bows....Do and undo, the day is long enough. We are likely to blunder less if we do not lose sight of our aim. The end of fishing is not angling but catching....The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what he has and uses it better. A wise man will make more opportunities than he finds. A wise man will make tools of what comes to hand. A wise man turns chance into good fortune. Or, possibly, the advantage of the expert is that he is continually on the lookout for opportunities. Have an eye to the main chance.
3. We should start carrying out our plan at the right moment, when it is ripe, but not before. We should not start rashly. Look before you leap. Try before you trust. A wise delay makes the road safe. On the other hand, we should not hesitate too long. If you will sail without danger, you must never put to sea. Do the likeliest and hope the best.....
We must use our judgment to determine the right moment. And here is a timely warning that points out the most common fallacy, the most usual failure of our judgment: We soon believe what we desire.
Our plan gives usually but a general outline. We have to convince ourselves that the details fit into the outline, and so we have to examine carefully each detail, one after the other. Step after step the ladder is ascended. Little by little as the cat ate the flickle. Do it by degrees.
In carrying out our plan we must be careful to arrange its steps in the proper order, which is frequently just the reverse of the order of invention. What a fool does at last, a wise man does at first.
4. Looking back at the completed solution is an important and instructive phase of the work. He thinks not well that thinks not again. Second thoughts are best.
Reexamining the solution, we may discover an additional confirmation of the result. ....such an additional confirmation is valuable, that two proofs are better than one. It is safe riding at two anchors.
5. ....Certain more systematic and more sophisticated aspects of the process of solution are hardly within the scope of the Wisdom of Proverbs. ....Here follow a few "synthetic" proverbs which describe somewhat more sophisticated attitudes.
The end suggests the means.
Your five best friends are What, Why, Where, When, and How. You ask What,, you ask Why, you ask Where, When and How --and ask nobody else when you need advice.
Do not believe anything but doubt only what is worth doubting.
look around when you have got your first mushroom or made your first discovery; they grow in clusters.
Source:
Polya, G. How to Solve It: A New Aspect of Mathematical Method. (Princeton, NJ: Princeton University Press, 1945, 2nd ed. 2004), pp. 222-225.
